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We have a total of 3+5+2 = 10 balls. If we take 6 balls, we have 10C6 total combinations, which equals 210.
Now, the fav combinations are best found the following way: we take each color separately and see how many ways there're to fill the given color's quota with different balls:
Red: 3C2 = 3
Blue: 5C2 = 10
Green: 2C2 = 1
Each combination within a given color we can mix with any other combinations of the two remaining color to get the total combination. For example, if red is filled with balls Red#1 and Red#2, we can mix it with Blue#1 and Blue#2, Blue#1 and Blue#3, etc. So, the total number of fav combinations is a multiplication of those numbers:
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